For requesting, clarifying, and comparing definitions of mathematical terms.

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65 views

Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
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114 views

Confused about the definitions of atom

I went though half a dozen books on measure theory, and it occurs to me that the definition of atom is not particularly unified. Version $1$: A set $E$ in a $\sigma$-algebra $\Sigma$ is called ...
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119 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
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1k views

Proper functions

I'm a bit confused with the definition of proper function, i.e.: "$f: X \to Y$ is proper if the inverse image of every compact set in $Y$ is compact in $X$." Can anyone provide a more intuitive ...
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59 views

Is the empty matrix $M$ ($m \times n$) with $m$ or $n$ equal to zero in RREF ? Does it posses every matrix property one would wish?

Is the empty matrix $M$ ($m \times n$) with $m$ or $n$ equal to zero in RREF ? Reading a proof in Linear Algebra induction starts off by proving $P(n)$ holds for $n=0$, where $P(n)$ is the ...
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393 views

What are defintions of ellipticity?

Ellipticity seems to have many definitions. So far, I'm aware of three. Are there other definitions of ellipticity that you know of, and if so, where did you encounter them? The three definitions ...
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111 views

Clarification of Hölder norm in terms of oscillation

Let $\Omega\subset\mathbb{R}^2$ be an open bounded set, $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$, $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$, $u\in L^{\infty}\big(\Omega(x, ...
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104 views

What is a Line Integral?

My question is very simple yet crucial to the understanding of many fields of mathematics. What is a line integral? If I choose some arbitrary line segment $\mathbb{A}$ to integrate a function ...
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118 views

Meaning of quasiperiodicity in classical KAM

I'm learning about the classical KAM theorem, and I can't quite infer precisely what the term "quasi-periodic solution" means in the theorem's statement. I'm reading the following introductory note: ...
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211 views

Are edge cuts, vertex cuts, and cut sets all variously called “cuts”?

I've seen "cut" being used to refer to all three, in different places, and sometimes in the same book. Which does "cut" most commonly refer to? p.s. I am aware that "cut" itself can be defined to ...
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343 views

Different Definitions on the Differentiability of Functions on a closed set.

I have encountered three different definitions on the differentiablity of functions on a closed set. In the following, suppose that $\Omega\subset M$ is a (open) domain, where $M$ is a manifold. The ...
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64 views

Can a sample space depend on the parameter to be estimated (example: # of cabs in a city)

In our intro statistic lecture the following we said that the following components made up an estimation problem an at most countable space $\mathcal{X}$ of all possible samples we can observe a ...
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93 views

Graph Theory: Help with a definition

I need some help to see if the definition I found of Cycles and Cycles Decomposition is right, here it is: A graph is a Cycle if it is isomorph to another graph $G=(V,E)$ with the following ...
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418 views

Explanation of Mixed Strategy Definition in Game Theory

Definition: Let $(N, A, u)$ be normal-form game, and for any set $X$ let $\Pi(X)$ be the set of all probability distributions over $X$. Then the set of fixed strategies for player $S_i=\Pi(A_i)$. ...
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84 views

Name for the initial object of a connected component of a category

I was just wondering whether there is already a name for an object of a category as described below. Recall that an initial object in a category $\mathcal C$ is an object $I\in\mathcal C$ such that ...
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94 views

Where can I find a description of math language symbols?

I am reading math articles. I meet math symbols. For example $\exists$ or $\forall$. For example for "For any a exist e that" can be rewriten as: $\forall a \exists e$ Where can I find full ...
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223 views

How to prove that $e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)} = \lim_{n \to \infty} \sqrt[n]{n\#} $?

While reading this post, I stumbled across these definitions (Wiki_german) $$e = \lim_{n \to \infty} \sqrt[n]{n\#}$$ and $$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}.$$ The last one seems ...
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57 views

terminology clarification needed: projection of a measure to an open set

I'm reading in Doobs "Classical Potential Theory and its Probabilistic Counterpart" and I'm having trouble with terminology. Specifically, in Part I chapter 6, he talks about the projection of a ...
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103 views

question about definition Silverman's AEC

I am sure many of you have Silverman's book the arithmetic of elliptic curves (2nd edition). On page 417, proposition 1.2, he defines $\delta$. He also defines $\xi: G \rightarrow M:\ ...
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283 views

why are curl and divergence defined the way they are?

Curl of a given vector field, $F$, at a given point is a vector, $C$, that gives the measure of amount of rotation the vector undergoes at that point. It has been defined using cross products and ...
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32 views

How to describe a polynomial relation on $\mathbb{P}(\bigwedge^k V)$, and if the Zariski topology is canonical

I am working with the space $\mathbb{P}(\bigwedge^k V)$, where $V$ is some $n$ dimensional vector space over some field K. In here I want to define a variety, ie a solution to a set of polynomials. ...
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20 views

Is a finite dimensional vector also a covector?

I am reading something where it seems they have defined a covector/linear functional as a map $F: \mathbb{R}^n \to \mathbb{R}$ where $F$ is a vector For example, $F(x) = \begin{bmatrix} 1 & 2 ...
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32 views

$\mathsf{Top}^2$ is a stack over $\mathsf{Top}$?

In section 4.1 of Vistoli's notes he starts by showing we may locally construct arrows in $\mathsf{Top}^2$, i.e arrows over $U$. Then, the author says that we can moreover do the same for spaces, ...
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41 views

different definitions of a group

A group is classically defined as a set with a binary operation (the group product) which is associative, such that there is a unit and for every element there is an inverse. I know we can define a ...
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47 views

How are neighbouring sequences defined? (Metric spaces)

What does it mean for sequences to be neighbours in a metric space? My attempt is: In a metric space $(X,d)$, $(x_n)$ and $(y_n)$ are neighbouring sequences iff $$\forall_{\epsilon>0}\exists_N ...
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34 views

Is there an accepted definition of coprimality in commutative ring theory?

I can think of at least three possible definitions of coprimality in commutative ring theory: $a,b \in R$ are coprime iff if $c \mid a$ and $c \mid b$, then $c \mid 1$. if $a \mid c$ and $b \mid c$, ...
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28 views

Short question about differential forms / exterior algebra

I am working towards understanding the wedge product of two vectors. Here is what I have so far: Let $V$ be an $n$-dimensional vector space over $\mathbb R$ (or $\mathbb C$, I'm not sure it makes ...
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29 views

Étale morphisms definition?

Working over a commutative ring $R$, let $D= \left\{ d\in R : d^2 =0 \right\}$ A formally étale morphism $f:M\rightarrow N$ is one for which the square below is a pullback for every point ...
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39 views

Is this how we define “limit in the distributional sense”?

Consider $\mathcal{D}(\mathbb{R})$ the space of test functions and $\mathcal{D}'(\mathbb{R})$ the space of distributions, in $\mathbb{R}$, i.e., continuous linear functionals over ...
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69 views

What is the name of the property $x^m=x$ when $x$ is in a ring?

I have been doing problems in Atiyah & MacDonald's Introduction to Commutative Algebra, and in problem 1.6 it asks to assume the existence of an idempotent element in an ideal whenever the ideal ...
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61 views

How are non-associative groups called?

Is there a name for the algebraic objects that have all the properties of groups expect associativity? For example, the unit octonions have this property. They satisfy the following definition. ...
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40 views

Does there exist a derivative of the Golden Ratio (equation)?

On my calculus exam, there was a question asking "Is the Golden Ratio differentiable? If so, use the definition of a derivative to show it." The "definition of a derivative" is ...
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15 views

Different meanings of $L_r(P)$: norm, metric, class of functions

I have doubts on the meaning of the symbol $L_r(P)$ for $r\in \mathbb{N}$. Consider a random variable $X: \Omega \rightarrow \mathbb{R}$ with probability distribution $P$ and a random function ...
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44 views

What does non-degenerate mean?

The context is that f is a real-valued function on a plane such that for every non-degenerate square ABCD in the plane, f(A)+f(B)+f(C)+f(D)=0.... What does this word mean here? I've found that "in ...
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39 views

Meaning of “Form” in mathematics

There are a number of occasions in the mathematics where the word "form" is used: Modular form Bilinear form Quadratic form ... It seems that form is a special kind of function. But I cannot ...
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19 views

Ideal Sheaf: Making sure I got it right

I want to know if I unwrapped the definition of ideal sheaves correctly. Let $\mathscr{O}$ be a sheaf of rings on topological space $X$. The definition of an ideal sheaf is as follows: The ideal ...
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39 views

What is the name of the sub-matrix?

Given a matrix $\mathbf{A}$ of size $n\times n$. Let $I=\{i_1,\ldots,i_k\}\subseteq\{1,\ldots,n\}$ for some $k\leqslant n$. How to call the sub-matrix of $\mathbf{A}$ that has its indices in $I$? (I ...
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18 views

Alternative to exponents

In a loose sense addition is repeated successorship, multiplication is repeated addition, and exponentiation is repeated multiplication. However, the latter is the only one that isn't well defined ...
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33 views

Quotient topology from Delta complex,

A $\Delta$-complex structure on a space X is a collection of maps $\sigma_{\alpha} : \Delta^n \rightarrow X$ with n depending on index $\alpha$ such that 1)The restriction ...
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54 views

If the sample space is an Euclidean Space, we can use a different type of PDF

The title resume all the point I'll try to make now. Reading this post, I realize that is possible to have another type of PDF (probability density function). Usually, we have a probability space ...
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20 views

Does every collection of edge vectors of a cone span a face of the cone?

Definition 1 : A polyhedral cone is a subset of a real vector space which is the intersection of finitely many closed half spaces. (The defining planes of these half spaces must pass through $0$.) A ...
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35 views

Definition of mathematical expression

According to wikipedia: "In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical ...
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41 views

Supply/demand digraph - understanding problem

I'm dealing with multiflows and I found in "Combinatorial Optimization - Part C" by Schrijver in Chapter 70 a good source. The definition of the multiflow problem involves so-called supply- and ...
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26 views

Set of 2-tuples with indices

I want to define the set of two tuples. Actually, I want to describe the degree distribution (or histogram) in terms of ordered tuples. The first item denotes the degree of a node. The second item ...
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29 views

Why was this terminology of holomorphism used here?

page 6 in this notesays: A note on index convention: $i$,$j$, and $\bar{i}$, $\bar{j}$ index over the holomorphic and antiholomorphic components. I never knew there were holomorphic ...
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27 views

Is this definition correct regarding the logic of recursive sequences?

This is a self study. I come across with a second order sequence of reals defined by: $$φ_{m+2}=φ_{⌊f(φ_{m},φ_{m+1})⌋}..................(*)$$ where $⌊.⌋$ is the integer part function. Here $f$ is a ...
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27 views

Definition of a local cross section

I came across the following definition of a cross section and local section in Wikipedia - Let $\pi : E\rightarrow B$ be a fibre bundle. A cross section of this bundle is defined as a continuous map ...
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29 views

Definition of derivative as best affine approximation

I'd like to try to redefine the derivative (in a way equivalent to the usual definition) of a function $f: U \subseteq R \to R$ to make it clear that the derivative $Df(a)$ is the linear part of the ...
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89 views

Definition: is a graph allowed to have a “dangling” edge without a vertex at its end(s)?

My textbook gives the following definition "a graph $G=(V,E)$ consisting of $V$, a nonempty set of vertices and $E$, a set of edges. Each edge has either one or two vertices associated with it." Now ...
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36 views

Can the parameter of prior probability depends on data?

In Bayseian approach https://en.wikipedia.org/wiki/Prior_probability we often use prior probability. Can we have a prior probability distribution with parameters and while estimating the posterior ...